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Properties

Label 5.1.20.27a
Base 5.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(27\)

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Defining polynomial

$x^{20} + 5 c_{10} x^{10} + 5 b_{9} x^{9} + 5 a_{8} x^{8} + 5 d_{0}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $27$
Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[\frac{1}{2}]$
Means: $\langle\frac{2}{5}\rangle$
Rams: $(2)$
Field count: $56$ (complete)
Ambiguity: $20$
Mass: $20$
Absolute Mass: $20$

Diagrams

Varying

Indices of inseparability: $[8,0]$
Associated inertia: $[1,1]$ (show 32), $[1,2]$ (show 8), $[1,4]$ (show 16)
Jump Set: undefined (show 42), $[1,13]$ (show 14)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 14

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
5.1.20.27a1.1 $x^{20} + 15 x^{8} + 5$ $C_5:C_4$ (as 20T2) $20$ $20$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.2 $x^{20} + 5 x^{10} + 15 x^{8} + 5$ not computed $ $not computed$ $ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.3 $x^{20} + 10 x^{10} + 15 x^{8} + 5$ not computed $ $not computed$ $ $10$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.4 $x^{20} + 5 x^{9} + 15 x^{8} + 5$ $C_5:F_5$ (as 20T26) $100$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.5 $x^{20} + 5 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.6 $x^{20} + 10 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.7 $x^{20} + 15 x^{10} + 5 x^{9} + 15 x^{8} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a1.8 $x^{20} + 20 x^{10} + 5 x^{9} + 15 x^{8} + 5$ not computed $ $not computed$ $ $5$ not computed not computed not computed not computed $[8, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 13]$
5.1.20.27a2.1 $x^{20} + 5 x^{8} + 5$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ $[1, 13]$
5.1.20.27a2.2 $x^{20} + 5 x^{9} + 5 x^{8} + 5$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 2$ $[1, 13]$
5.1.20.27a3.1 $x^{20} + 20 x^{8} + 5$ $C_4\times F_5$ (as 20T20) $80$ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ $[1, 13]$
5.1.20.27a3.2 $x^{20} + 5 x^{9} + 20 x^{8} + 5$ $F_5^2$ (as 20T102) $400$ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 4]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 3$ $[1, 13]$
5.1.20.27a4.1 $x^{20} + 10 x^{8} + 5$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ $[1, 13]$
5.1.20.27a4.2 $x^{20} + 5 x^{9} + 10 x^{8} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 0]$ $[1, 2]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 4$ $[1, 13]$
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